Find the value of term 1 given S8 is equal to -3280 and the common ratio is equal to -3?
Ah. Finally!
a(1-3^8)/(1+3) = -3280
-1640a = -3280
a = 2
Well, finding term 1 is like finding the first byte of a funny joke - it's the starting point that sets the tone! Now, using our laughter-infused math wizardry, we can use the formula for the sum of a geometric series to find term 1. The formula is S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, we're given S8 as -3280 and r as -3. Plugging in the values, we have -3280 = a(1 - (-3)^8) / (1 - (-3)). Now, if we solve this equation, we'll get the value of term 1. Just be patient - solving it might take longer than waiting for your friend to finish telling a knock-knock joke.
To find the value of term 1, we can use the formula for the sum of a geometric series:
S_n = a(1 - r^n) / (1 - r)
Where:
S_n: Sum of the first n terms of the geometric series
a: First term of the geometric series
r: Common ratio of the geometric series
n: Number of terms in the geometric series
From the given information, we have S8 = -3280 and r = -3. Let's substitute these values into the formula and solve for a:
-3280 = a(1 - (-3)^8) / (1 - (-3))
Simplifying:
-3280 = a(1 - 6561) / 4
-3280 = a(-6560) / 4
-4 * 3280 = a(-6560)
-13120 = -6560a
a = -13120 / -6560
a = 2
Therefore, the value of term 1 is 2.
To find the value of term 1 (a1) given S8, which represents the sum of the first 8 terms, and a common ratio of -3, we can use the formula for the sum of a geometric series:
S(n) = a1 * (1 - r^n) / (1 - r)
where S(n) is the sum of the first n terms, a1 is the first term, r is the common ratio, and n is the number of terms.
Now we have the value of S8 as -3280 and the common ratio as -3, so we can substitute these values into the formula:
-3280 = a1 * (1 - (-3)^8) / (1 - (-3))
Next, we can simplify the formula:
-3280 = a1 * (1 - 6561) / 4
-3280 = a1 * (-6559) / 4
To solve for a1, let's multiply both sides of the equation by 4:
-13120 = a1 * (-6559)
Then, divide both sides by -6559:
a1 = -13120 / (-6559)
Calculating this division, we get:
a1 ≈ 1.997
Therefore, the value of term 1 (a1) is approximately 1.997.